Question

The equation of the tangent to the parabola $y^2=9x$ which passes through the point $(4,10)$ is

• A. $x+4y+1=0$
• B. $x-4y+36=0$
• C. $9x-4y+4=0$
• D. $9x+4y+4=0$

Answer 1

Answer: (B), (C)

The correct options are
B $x-4y+36=0$
C $9x-4y+4=0$

Equation of a tangent to a parabola is given by,

$y=mx+\frac{a}{m}$ ------ ( 1 )

In the given parabola, $y^2=9x$

$\Rightarrow$ $4a=9$

$\Rightarrow$ $a=\frac{9}{4}$

Putting value of $a$ in ( 1 ) we get,

$y=mx+\frac{9}{4x}$ ----- ( 2 )

As the given tangent passing through point $(4,10)$

$\therefore$ $10=4m+\frac{9}{4m}$ ----- ( 2 )

$\Rightarrow$ $16m^2-40m+9=0$

$\Rightarrow$ $16m^2-36m-4m+9=0$

$\Rightarrow$ $4m(4m-9)-1(4m-9)=0$

$\Rightarrow$ $(4m-9)(4m-\frac{1}{4})=0$

$\Rightarrow$ $m=\frac{9}{4}$ and $m=\frac{1}{4}$

When $m=\frac{9}{4}$,

$\Rightarrow$ $y=\frac{9}{4}x+1$

$\Rightarrow$ $4y=9x+4$

$\Rightarrow$ $9x-4y+4=0$

When $m=\frac{1}{4},$

$\Rightarrow$ $y=\frac{1}{4}x+9$

$\Rightarrow$ $4y=x+36$

$\Rightarrow$ $x-4y+36=0$

$\therefore$ The required equations are $9x-4y+4=0$ and $x-4y+36=0$